If parallel forces \(P_1, P_2, \dots\) act at points \((x_1y_1), (x_2y_2), \dots\) of a plane, show that their resultant acts at the point \(\Sigma Px / \Sigma P, \Sigma Py / \Sigma P\). A uniform rectangular table ABCD is supported on three legs, two of which are at A, B. A third leg is attached at a point P so that the pressure at A is \(n\) times the pressure at B. Prove that P lies on a certain straight line and find the portion of the line for which the pressures are all positive.
A rough plank of thickness \(2b\) is laid across a fixed cylinder of radius \(a\) and rests in equilibrium at an angle \(\alpha\) with the horizontal. If the plank is rolled through an angle \(\theta\), find the increase in its potential energy and deduce that the equilibrium is stable if \(b < a \cos^2 \alpha\).
One end of a string is fixed, and the string, hanging in two vertical portions on the loop of which a ring of mass \(m\) moves, passes over a fixed pulley and has a mass \(M\) suspended from the other end. Show that when the system moves the pull of the string on its support is \[ \frac{3Mm}{4M+m}g. \]
Find the Cartesian equation of the path of a particle projected from the origin with component velocities \(u, v\) along the axes, the \(y\) axis being taken vertically upwards. If a particle is projected so as just to clear a wall of height \(b\) at a horizontal distance \(a\) and to have a range \(c\) from the point of projection, show that the velocity of projection (\(V\)) is given by \[ \frac{2V^2}{g} = \frac{a^2(c-a)^2+b^2c^2}{ab(c-a)}. \]
Two equal smooth spheres moving along parallel lines in opposite directions with velocities \(u, v\) collide with the line of centres at an angle \(\alpha\) with the direction of motion. If after impact their lines of motion are at right angles to one another, show that \[ \left(\frac{u-v}{u+v}\right)^2 = \sin^2\alpha + e^2\cos^2\alpha, \] where \(e\) is the coefficient of elasticity.
Find the period of the small oscillations of a simple pendulum. If a clock is moving horizontally with uniform acceleration in the plane of motion of its pendulum and in a true interval \(t\) indicates the lapse of an interval \(t'\), find the acceleration of the clock.
A wheel is kept revolving uniformly about a horizontal axis \(1\frac{1}{2}\) inches from its centre of gravity. Find the number of revolutions per minute if the pressure on the axis varies from one half to one and a half times the weight of the wheel. (Take \(\pi^2=10\).) Show that the greatest inclination to the vertical of the pressure of the wheel on its axis is 30\(^\circ\).
Solve the equation
Find the conditions that \(ax^2+2bx+c\) may be positive for all real values of \(x\). Shew that the expression \(\frac{8x}{x+a} + \frac{18x}{x-a}\) cannot have values lying between 1 and 25.
Solution: We need \(a \geq 0\) so that it's positive when \(x \to \infty\). If \(a = 0\) then it's linear so \(b = 0, c > 0\) We need \(\Delta = (-2b)^2 - 4ac = 4(b^2-ac) <0\) therefore \(b^2 < ac\). Therefore the conditions are \(a > 0, b^2 < ac\) or \(a = b = 0, c > 0\) \begin{align*} && \frac{8x}{x+a} + \frac{18x}{x-a} &= \frac{8x^2-8ax+18x^2+18ax}{x^2-a^2} \\ &&&= \frac{26x^2+10ax}{x^2-a^2} \\ && k &= \frac{26x^2+10ax}{x^2-a^2} \\ && 0 &= (26-k)x^2+10ax+ka^2 \end{align*} this is always positive if \(k < 26\) and \begin{align*} && 25a^2 &< (26-k)ka^2 \\ && 25 &< (26-k)k \\ && 0 &< -k^2+26k-25 \\ &&&= -(k-25)(k-1) \end{align*} Therefore \(k \in (1,25)\) since all these values have \(k < 26\) too.
Find an expression giving all the angles which have the same sine as \(A\). Solve the equation \begin{align*} &\cos x \sin(x-a)\sin(x-b)\sin(x-c) \\ &+ \sin x \cos(x-a)\cos(x-b)\cos(x-c) = \sin a \sin b \sin c. \end{align*}